Verified Implementation of Exact Real Arithmetic in Type Theory

Transcription

1 Verified Implementation of Exact Real Arithmetic in Type Theory Bas Spitters EU STREP-FET ForMath Dec 9th 2013 Computable Analysis and Rigorous Numerics 1 / 14

2 Rigorous Numerics through Computable Analysis Constructive analysis as an internal language for TTE Type theory as a language for constructive mathematics Type theory as a framework for computer proofs Computer verified implementation of exact analysis 2 / 14

3 Dependent type theory Dependent type theory makes Bishop s notion of operation/construction precise. Gives a functional programming language like haskell, SML, OCaml with very expressive type system. Framework for proofs Implementations: Coq, agda, epigram, Idris,... Extensional DTT is an abstract language for TTE via realizability. Locally Cartesian closed category, ΠW-pretopos. 3 / 14

4 Picard-Lindelöf Theorem Consider the initial value problem where y (x) = v(x, y(x)), y(x 0 ) = y 0 v : [x 0 a, x 0 + a] [y 0 K, y 0 + K] R y v is continuous y v is Lipschitz continuous in y: 0 + K v(x, y) v(x, y ) L y y for some L > 0 y 0 v(x, y) M al < 1 y 0 K am K x 0 a Such problem has a unique solution on [x 0 a, x 0 + a]. x 0 x 0 + a x 4 / 14

5 Proof Idea is equivalent to Define y (x) = v(x, y(x)), y(x 0 ) = y 0 x y(x) = y(x 0 ) + v(t, y(t)) dt x 0 x (T f)(x) = y 0 + F (t, f(t)) dt x 0 f 0 (x) = y 0 f n+1 = T f n Under the assumptions, T is a contraction on C([x 0 a, x 0 + a], [y 0 K, y 0 + K]). By the Banach fixpoint theorem, T has a fixpoint f and f n f. Formalization: Makarov, S - The Picard Algorithm for Ordinary Differential Equations in Coq 5 / 14

6 Metric Spaces Let (X, d) where d : X X R be a metric space. Let Brxy denote d(x, y) r. A function f : Q + X is called regular (Strongly Cauchy) if ε 1 ε 2 : Q +, B(ε 1 + ε 2 )(fε 1 )(fε 2 ). The completion C X of X is the set of regular functions. Let X and Y be metric spaces. A function f : X Y is called uniformly continuous with modulus µ if ε : Q + x 1 x 2 : X, B(µε)x 1 x 2 Bε(fx 1 )(fx 2 ). For x 1, x 2 : C X, the metric on the completion B C X εx 1 x 2 := ε 1 ε 2 : Q +, B X (ε 1 + ε + ε 2 )(x 1 ε 1 )(x 2 ε 2 ). Metric spaces with uniformly continuous functions form a category. Completion forms a monad in the category of metric spaces and uniformly continuous functions. R. O Connor, extending work by E. Bishop. 6 / 14

7 Completion as a Monad unit : X C X := λxλε, x join : C C X C X := λxλε, x(ε/2)(ε/2) map : (X Y ) (C X C Y ) := λfλx, f x µ f bind : (X C Y ) (C X C Y ) := join map Define functions Q Q; lift to C Q C Q. 7 / 14

8 Type Classes Organize the library using, so-called type classes. Type classes are parametric record types. Coq searches for terms of these types automatically during unification. Logic programming at the type level automates many mathematical reflexes. Gives uniform notation Algebra hierarchy, abstractions, diamond inheritance Abstract interfaces (like haskell). S, van der Weegen, Type Classes for Mathematics in Type Theory. 8 / 14

9 Type Classes for Mathematical Structures Class AppRationals AQ {e plus mult zero one inv} {Apart AQ} {Le AQ} {Lt AQ} {AQtoQ : Cast AQ Q_as_MetricSpace} {!AppInverse AQtoQ} {ZtoAQ : Cast Z AQ} {!AppDiv AQ} {!AppApprox AQ} {!Abs AQ} {!Pow AQ N} {!ShiftL AQ Z} { x y : AQ, Decision (x = y)} { x y : AQ, Decision (x y)} : Prop := { aq_ring AQ e plus mult zero one inv ; aq_trivial_apart :> TrivialApart AQ ; aq_order_embed :> OrderEmbedding AQtoQ ; aq_strict_order_embed :> StrictOrderEmbedding AQtoQ ; aq_ring_morphism :> SemiRing_Morphism AQtoQ ; aq_dense_embedding :> DenseEmbedding AQtoQ ; aq_div : x y k, ball (2 ^ k) ( app_div x y k) ( x / y) ; aq_compress : x k, ball (2 ^ k) ( app_approx x k) ( x) ; aq_shift :> ShiftLSpec AQ Z ( ) ; aq_nat_pow :> NatPowSpec AQ N (^) ; aq_ints_mor :> SemiRing_Morphism ZtoAQ }. S, Krebbers, Type classes for efficient exact real arithmetic in Coq 9 / 14

10 Instances of Approximate Rationals Record Dyadic Z := dyadic { mant: Z; expo: Z }. Represents mant 2 expo Instance dy_mult: Mult Dyadic := λ x y, dyadic (mant x * mant y) (expo x + expo y). Instance : AppRationals (Dyadic bigz). Instance : AppRationals bigq. Instance : AppRationals Q. Waiting for MPFR/Coq interval/floqc / 14

11 Efficient Reals Coq < Check Complete. Complete : MetricSpace -> MetricSpace Coq < Check Q_as_MetricSpace. Q_as_MetricSpace : MetricSpace Coq < Check AQ_as_MetricSpace. AQ_as_MetricSpace : (AQ : Type)..., AppRationals AQ -> MetricSpace Coq < Definition CR := Complete Q_as_MetricSpace. Coq < Definition AR := Complete AQ_as_MetricSpace. AR is an instance of Le, Field, SemiRingOrder, etc., from the MathClasses library. 11 / 14

12 Integral Following M. Bridger, Real Analysis: A Constructive Approach. Class Integral (f: Q -> CR) := integrate: forall (from: Q) (w: QnonNeg), CR. Notation " " := integrate. Class Integrable {!Integral f}: Prop := { integral_additive: forall (a: Q) b c, f a b + f (a + b) c == f a (b + c); integral_bounded_prim: forall (from: Q) (width: Qpos) (mid: Q) ( (forall x, from <= x <= from + width -> ball r (f x) mid) -> ball (width * r) ( f from width) (width * mid); }. Earlier (abstract, but slower) implementation of integral by O Connor and S 12 / 14

13 Complexity Rectangle rule: b f(x) dx f(a)(b a) a where f (x) M for a x b. (b a)3 M 24 Number of intervals to have the error ε: Simpson s rule: b f(x) dx b a 6 a ( f(a) + 4f where f (4) (x) M for a x b. ( a + b Number of intervals: 4 (b a) 5 M 2880ε 2 (b a) 3 M 24ε ) ) (b a)5 + f(b) 2880 M Coquand, S A constructive proof of Simpson s Rule, 2012: Replace mean value theorem with law of bounded change Use divided differences, Hermite-Genocci 13 / 14

14 Conclusions Computer verified implementation of simple ODE solver. Computing with exact functions. May be seen as an executable specification, speed up with refinement. 14 / 14